The last integral over \(f_{11}\)  can be explicitly evaluated by using the formula 3.382.7 of [1]}:
\(\int e^{-ifp}\frac{1}{(\beta -if)^{\nu }}\frac{df}{2\pi }=\frac{p^{\nu -1}}{\Gamma {(\nu )}}e^{-\beta p}\theta (p), \quad Re(\nu ,\beta )>0\) 
